steady state error PnPn

[Philipp Schlatter]

dear all,

For a stability problem where we want to calculate the base flow with
simple time marching. We found some discrepancies in our case (a helical
pipe flow) with results from a fully spectral 2D code. Not much but
something a bit larger than the tolerances. Since we could not think of
further reasons, we were wondering whether they could be any explanation
lying in the method itself. Does the PnPn method perhaps have some
steady state error, or should it, in principle, converge to zero error
independent of time step? I think it should, but perhaps I miss something?

Thanks for any hint!

Best regards,
Philipp

[YuHsiang Lan]

Hi Philipp,

What boundary conditions are you using?

The numerical divergence (from PnPn) is governed by the diffusion, or (modified) Helmholtz after discretization. 

Therefore, the div err should decay to zero in time as long as the BC is exact.

There is a du/dt BC term at inlet, so If your BC is time dependent, it can introduce O(dt ^ nEXT) error.

One easy mistake I ran into recently is prescribing a constant inflow without adjusting the scaling for the inlet-wall corner.

Since we enforce zero velocity at the wall, the effective inflow rate becomes smaller than intended.

A rough estimate:

- N-th order

- Number of edges at inlet-wall corner is O(1)

- Number of grid points along the inlet-wall edges is O(N)

- 1D GLL quadrature weights at end point: 2/[N*(N+1)] = O(N^{-2}), assuming length is also O(1)

  This is also the averaged 2D weights along that line because sum_j (w_j) = 2.

Overall, this leads to an O(1/N) loss for the flowrate, which ruins the exponential convergence.

Hope this helps,

Yu-Hsiang

--

[Philipp Schlatter]

Thanks!

In our case, it is periodic (or rather, cyclic) and no-slip a the pipe walls, so in that sense the boundary conditions are "excact". I was just wondering whether there could be some sort of steady-state error, as it is for plan1 or plan2 (I forgot...).

Best, Philipp

--
You received this message because you are subscribed to the Google Groups "Nek5000" group.
To unsubscribe from this group and stop receiving emails from it, send an email to nek5000+u...@googlegroups.com.
To view this discussion visit https://groups.google.com/d/msgid/nek5000/d34108cb-69ba-4c98-81e0-24b2ce7a031dn%40googlegroups.com.